The correlation between fragility, density and atomic interaction in

The correlation between fragility, density and atomic interaction in glassforming liquids
Lijin Wang1, Pengfei Guan1* , and W. H. Wang2
1
Beijing Computational Science Research Center, Beijing, 100193, P. R. China.
2
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, P. R. China
*
Corresponding E-mail: [email protected]
The fragility that controls the temperature-dependent viscous properties of liquids as the glass transition is
approached, in various glass-forming liquids with different softness of the repulsive part of atomic interactions at
different densities is investigated by molecular dynamic simulations. We show the landscape of fragility in purely
repulsive systems can be separated into three regions denoted as RI, RII and RIII, respectively, with qualitatively
disparate dynamic behaviors: RI which can be described by β€˜softness makes strong glasses’, RII where fragility is
independent of softness and can only be tuned by density, and RIII with constant fragility, suggesting that density plays an
unexpected role for understanding the repulsive softness dependence of fragility. What more important is that we
unify the long-standing inconsistence with respect to the repulsive softness dependence of fragility by observing that
a glass former can be tuned more fragile if nonperturbative attraction is added into it. Moreover, we find that the
vastly dissimilar influences of attractive interaction on fragility could be estimated from the structural properties of
related zero-temperature glasses.
leading to the ultimate understanding of glass transition,
remains unresolved.
I. INTRODUCTION
Recently, fragility has been investigated in a series of
experimental and theoretical studies 15-21 by taking into
consideration the details of constituent interatomic
potentials which underpin many properties of glasses and
glass formers. 22-25 Apparent correlations between
fragility and specific properties of interaction potential
have been observed. However, some of them beyond our
expectations have so far proved inconclusive or contrary.
One crucial but inconclusive question up to now is how
softness of repulsion affects fragility. In the colloidal
experiment, 15 it is claimed that softness makes strong
glasses, while different conclusions have also been
drawn, e.g. softness increases fragility in modified
Lennard-Jonnes systems, 16-19 and fragility is independent
of softness in inverse power law potential systems. 20
In this paper, a series of glass forming systems
composing of particles with various densities, interacting
via modified Lennard-Jonnes potentials with different
steepness of the repulsive interaction, were studied to
investigate the correlation between the fragility, density
and atomic interaction. We unified the long-standing
inconsistence15-20 with respect to the repulsive softness
dependence of fragility. It is found that the landscape of
fragility in purely repulsive systems can be separated into
three regions with qualitatively disparate dynamic
behaviours. Furthermore, the influence of attractive
Understanding the nature of glass transition is still a
major open question in condensed matter physics, 1-6
though great efforts have been made over the years. An
important aspect of this question is elucidating how the
structural relaxation time 𝜏 of a glass-forming liquid
increases significantly on cooling and drives the system
towards its glassy state at some temperature 𝑇𝑔 . The
kinetic fragility index7-9 π‘š which is usually defined by
π‘š=
𝑑 π‘™π‘œπ‘” 𝜏
𝑑(
|
𝑇𝑔
)
𝑇
𝑇=𝑇𝑔
, characterizes how rapidly the π‘™π‘œπ‘”(𝜏)
changes with 𝑇𝑔 /𝑇 at 𝑇 = 𝑇𝑔 and presents the singular
dynamic behaviours of different glass-forming liquids. 𝑇𝑔
is the glass transition temperature below which the
material has become so viscous that its typical relaxation
time scale (of the order of 100s)5 exceeds the measurable
time window. The fragility π‘š exhibits great diversity
over a wide swath of glass formers from colloids 3 to
atoms5, and the liquids can be defined as strong or fragile
liquids with small or large π‘š, respectively. Despite the
fact that numerous empirical correlations between
fragility and other dynamical, 10,11 thermo-dynamical,
9,10,12
vibrational13 or mechanical14 properties have been
observed in different glass formers, the physical origin of
fragility, deemed to be one of the key issues likely
1
FIG. 2. (a): Temperature 𝑇 dependence of the selfintermediate scattering function 𝐹s (q, t) of glass forming
liquids at density, 𝜌 = 1.2 for model π’ŒπŸ’rlj without
attraction. Relaxation time 𝜏 at each 𝑇 is determined
from 𝐹s (q, t) = eβˆ’1 which is marked with horizontal
dashed line. (b): T dependence of 𝜏. The solid line is fit
to VFT function. The horizontal dashed line marks 𝜏 =
106 at which glass transition temperature 𝑇g is extracted.
FIG. 1. Schematic illustration of the scaled LennardJones potential 𝑉(π‘Ÿij )/πœ–ij . Inset: Density 𝜌 dependence of
the position βŒ©π‘ŸβŒͺ, as illustrated in the sub-inset, of the first
peak of pair distribution function of A particles 𝑔𝐴𝐴 (π‘Ÿ)
of inherent structures for model π’ŒπŸ’rlj . The solid line in
the inset is the fit according to βŒ©π‘ŸβŒͺ ~ πœŒβˆ’1/3 . The vertical
dashed lines from right to left mark βŒ©π‘ŸβŒͺ at ρ = 1.2, 1.4,
1.6 and 1.8, respectively. The purple arrow locates the
minimum of 𝑉(π‘Ÿij )/πœ–ij .
are chosen respectively for Lennard-Jonnes (LJ) system
with both attraction and repulsion, and purely Repulsive
Lennard-Jonnes (RLJ) system which is the same as the
Weeks-Chandler-Andersen potential used in Refs. 24 and
25.
To suppress crystallization, we employ in our
simulations the extensively used parameters proposed by
Kob and Anderson in the standard Lennard-Jonnes
potential26 which is our model k4(12,6): πœ–AB = 1.5πœ–AA ,
πœ–BB = 0.5πœ–AA , 𝜎AB = 0.8𝜎AA , and 𝜎BB = 0.88𝜎AA . And
indeed no crystallization is observed in our simulation.
The units for length, energy and mass scales are set to be
𝜎AA , πœ–AA and 𝑀, respectively, and hence time is in units
2
of (π‘€πœŽAA
/πœ–AA )1/2 . Temperature 𝑇 is expressed in units
of πœ–π΄π΄ /π‘˜π΅ with π‘˜π΅ the Boltzmann constant. We also
performed simulations in k4(12,6) model with other
parameters as studied in Ref. 6 to make sure that our
conclusions drawn from this study won’t be affected by
the choice of parameters for avoiding crystallization.
The fast inertial relaxation engine minimization
algorithm28 is employed to generate zero-temperature
glasses (also termed inherent structures 9) by quenching
an initial high-temperature equilibrium state to its
corresponding local potential energy minimum. The
structure of inherent structures is measured by the pair
distribution function of A particles, 𝑔AA (π‘Ÿ) =
𝐿3
< βˆ‘π‘– βˆ‘π‘—β‰ π‘– Ξ΄(rβˆ’rij )> , where 𝑁𝐴 is the number of A
𝑁 2
interaction on fragility can be estimated from the
structural properties of related zero-temperature glass.
II. SIMULATION DETAILS
Our three dimensional cubic simulation box of side
length 𝐿 with periodic boundary conditions applied in all
directions is composed of N = 1000 (800 A and 200 B)
particles with equal mass M. Density (𝜌) is defined as ρ=
N/L3. Particles i and j, when their separation π‘Ÿij is less
than the potential cutoff π‘Ÿij𝑐 , interact via a family of (q, p)
Lennard-Jones
potentials
having the
following
generalized form: 16,26
𝑉(π‘Ÿπ‘–π‘— ) =
π‘Ÿij0
πœ–π‘–π‘—
π‘žβˆ’π‘
π‘Ÿ0
π‘ž
[𝑝 ( 𝑖𝑗 ) βˆ’ π‘ž (
π‘Ÿπ‘–π‘—
0
π‘Ÿπ‘–π‘—
π‘Ÿπ‘–π‘—
𝑝
) )] + 𝑓(π‘Ÿπ‘–π‘— ), (1)
1/6
where
= 2 πœŽπ‘–π‘— is the position of the minimum of
𝑉(π‘Ÿπ‘–π‘— ) and the correction function f(rij) is added to satisfy
continuity of V(rij) and its first derivative V0(rij) at π‘Ÿij𝑐 , 27
i.e. V( π‘Ÿij𝑐 ) = V0( π‘Ÿij𝑐 ) = 0. Obviously, the steepness of
repulsive part (k) can be tuned by various combinations
of (q, p), as shown in Fig. 1. We studied systems with six
combinations of (q, p) which are denoted in ascending
order by steepness of repulsion as k1(2,1), k2(6,4),
k3(8,5), k4(12,6), k5(12,11) and k6(18,16), respectively.
For consistency and comparison with previous studies1619
in the same generalized Lennard-Jonnes potential
systems, we will use denotations from k1 to k6 to
characterize different steepness of the repulsive part of
potential. Potential cutoffs π‘Ÿij𝑐 = 2.5πœŽπ‘–π‘— and π‘Ÿij𝑐 = 21/6 πœŽπ‘–π‘—
𝐴
particles, the sums are over all A particles and < . >
denotes the average over different inherent structures.
For glass-forming liquids, molecular dynamics
simulations are carried out in the canonical ensemble
with a 4-variable Gear predictor-corrector integrator 27
used to propagate the motion of each particle. Data won't
be collected until the system has been equilibrated for
several structural relaxation time 𝜏 determined by the
time when the self-intermediate scattering function of A
2
curves can be observed as the k increases from k4rlj to
k6rlj in Fig. 3(a), which suggests that there is an onset
steepness kc above which π‘š becomes insensitive to the k.
It is found that kc could be a function of 𝜌 and kc(𝜌 )
decreases with larger density as evidenced in Fig. 3(b)
for 𝜌 = 2.0 where curves start to collapse at about k2rlj. It
demonstrates that the steepness dependence of fragility at
fixed density exhibits disparate behaviours at different
domains of steepness, and that softness makes strong
glasses applies only in the context of a specific steepness
range with k weaker than kc(𝜌). On the other hand, we
also investigated the density-dependence of fragility for
each steepness k. Fig. 3(c) shows the density dependence
of fragility for k4rlj, and the fragility increases with larger
density, consistent with observations in other repulsive
systems,8,30 but attains constant above an onset density
πœŒπ‘ , which, due to limited range of densities in previous
studies, has never been observed before in RLJ systems.
As shown in the inset to Fig. 1, the inter-particle distance
βŒ©π‘ŸβŒͺ characterized by the first peak position of pair
distribution functions, is a monotone decreasing function
of density of this form, βŒ©π‘ŸβŒͺ ~ πœŒβˆ’1/3 . It means the region
of steepness sampled by particles varies with 𝜌 for k4rlj.
Therefore, fragility's density dependence at fixed
steepness model is, to a certain degree, analogous to its
steepness dependence at fixed density. The onset density
πœŒπ‘ can be extracted for each fixed k model and the
invariant π‘š can be obtained for 𝜌 > 𝜌c(k) cases.
Therefore, we enlarged our view and considered the
entire 𝜌 – k field for RLJ systems. Fig. 4 shows the
fragility π‘š as a two-dimensional function of 𝜌 and k. We
FIG. 3. (a), (b) and (c): Angell plot of relaxation time 𝜏
versus scaled reciprocal temperature 𝑇g /𝑇 in RLJ
systems. Densities in panels (a) and (b) are 1.2 and 2.0,
respectively, and symbols in panel (b) have the same
implications as in (a). Data demonstrated in panel (c) is
from model k4rlj. Through data points in panels (a), (b)
and (c) are VFT fit lines. (d): Scaling collapse of pair
distribution functions of A particles gAA(r) at different
densities in model k4rlj.
particles, 𝐹s (π‘ž, 𝑑) =
1
𝑁𝐴
< βˆ‘π‘— exp(π‘–π‘ž
βƒ— βˆ™ [π‘Ÿπ‘— (𝑑) βˆ’ π‘Ÿπ‘— (0)]) > ,
with π‘Ÿπ‘— (𝑑) the location of particle j at time 𝑑, π‘ž
βƒ— chosen in
the y direction with the amplitude approximately equal to
the position of the first peak of the static structure
factor,27 the sum being taken over all A particles and <
. > denoting time average, decays to 𝑒 βˆ’1 .8 The glass
transition temperature 𝑇g defined as 𝜏(𝑇g ) = 106 in our
study is extracted by fitting 𝜏 using Vogel-Fulcher𝐢
Tamman (VFT) function, 29 𝜏 = 𝜏0 exp(π‘‡βˆ’π‘‡
) , where 𝑇0 ,
0
𝜏0 and C, though having their separate physical
significance,19 are used here simply as fitting parameters.
Fig. 2 illustrates how we determine 𝜏 from 𝐹s (π‘ž, 𝑑) and
extract 𝑇g through VFT fit.
III. RESULTS AND DISCUSSION
Typical Angell plots of 𝜏 as a function of Tg/T within
RLJ system are shown in Fig. 3(a), (b) and (c) for
various densities and all data can be well fitted by VFT
function. Attention will be first focused on Fig. 3(a)
which compares Angell plots between models from k1rlj
to k6rlj at the widely used density, 26 𝜌 = 1.2. Seen clearly
from k1rlj to k4rlj, the curve becomes steeper with
increasing steepness, indicating that fragility π‘š increases
with enhanced steepness, i.e. softness makes strong glass
formers, consistent with the colloidal experiment
observation.15 Then, an approximate collapse of different
FIG. 4. Schematic contour plot of fragility ( π‘š ) as a
function of density (𝜌) for different steepness models (π’Œ)
in RLJ systems. The crosses in the panel indicate all the
data points we have obtained in our study and the circles
mark the locations of the four models which will be
discussed in detail in Fig. 5. More detailed description of
the plot can be found in the text.
3
found that the fragility in the 𝜌 – k field for RLJ systems
can be divided into three regions with qualitatively
distinct behaviours. The white solid line which is the
upper boundary of Region I (denoted as RI) is
determined by the function of density-dependent onset
steepness, k = kc( 𝜌 ). The conclusion about β€˜softness
makes strong glasses’ can only be achieved in RI and
fragility becomes insensitive to the k for models in
Region II (denoted as RII). The fragility in RII can only
be tuned by 𝜌, which suggests the profile of potential
energy landscape (PEL) 9 of systems belonging to RII is
mainly dominated by particle density. The lower
boundary of Region III (denoted as RIII) is described by
the function of steepness-dependent onset density, 𝜌 =
πœŒπ‘ (π’Œ)
and
the models in
RIII
present
uncontrollable constant fragility. It implies that the
systems in RIII presenting identical dynamic evolution
behaviours near Tg may provide us a good platform to
understand the glass transition.
For a serial of systems with same k but different 𝜌, the
density-dependent dynamic behaviors near 𝑇𝑔 can be
understood by the structural properties of related zerotemperature glassy states. The pair distribution functions
of A particles g AA (r) against π‘ŸπœŒ1/3 are shown in Fig.
3(d) for k4rlj. It presents nice scaling collapse above a
crossover density which almost coincides with πœŒπ‘ (π’Œ)
( πœŒπ‘ (π’Œ) = 1.8 for k4rlj as shown in Fig. 3(c)). Similar
scaling collapse has recently been observed in LJ
glasses24 and liquids31. We find the similarities of
structures of inherent structures above the steepnessdependent crossover density can be observed in all the
six steepness models in our study. Even for the softest
model k1rlj, the heights of the first peak of 𝑔AA (π‘Ÿ) almost
don't vary with densities for 𝜌> 2.0 (not shown here).
Therefore, for one certain k, the invariant fragility due to
the high density effect can be explained by the isomorph
of inherent structures. However, it should be mentioned
that neither isomorphic structures nor similarities of
structures can be observed to explain the hard steepness
effect at fixed density. It suggests that the two-pointcorrelation function 𝑔(r) can’t provide a full
understanding of dynamic behaviors and hence highorder correlation information should be included.
So far we are concerned with steepness dependence of
fragility in purely repulsive systems. Generally, attractive
interactions are usually treated as perturbation in the
theory of liquids.32 However, it has been recently
demonstrated that glass formers with attraction exhibit
more sluggish25 and heterogeneous23 dynamics than those
with pure repulsion, and hence attraction doesn’t act as
perturbation. Though the non-perturbative effect of
attraction on some quantities23-25 under certain
conditions24 has been put forward for years, to our
knowledge, no attention has yet been focused on probing
the attraction dependence of fragility. In order to probe
whether and how attraction affects fragility, we chose
four RLJ models (marked as A, B, C, and D) which are
FIG. 5. (a) and (b): Angell plot of relaxation time 𝜏
versus scaled reciprocal temperature 𝑇g /𝑇 for models
(𝜌, π’Œ) = (1.2, k3), (1.2, k5), (1.2, k4) and (1.9, k4) which
are marked as A, B, C and D in Fig. 4, respectively.
Through data points in panels (a) and (b) are VFT fit
lines. (c): Steepness k dependence of fragility index π‘š in
RLJ and LJ systems at fixed density, 𝜌 = 1.2. The two
regions I and II with different colors are subsets of RI
and RII in Fig. 4, respectively. (d): Comparison of
Δ𝑔AA (π‘Ÿ) as a function of π‘Ÿ between models A, B, C and
D.
located on different regions as shown in Fig. 4, and
introduced the attraction into the atomic interaction. The
calculated Angell plots are shown in Fig. 5 (a) and (b),
and the data of related RLJ systems are also present for
comparison. The divergences between the plots of the
models (A, B and C) with and without attraction indicate
that the attraction in the systems belonging to RI can’t be
treated as perturbation and makes them more fragile than
respective RLJ models. As shown in Fig. 5(b), the
collapse of the two curves of model D (with and without
attraction, respectively) indicates that the attraction effect
in RIII can be avoided and we can employ the RLJ model
to investigate the dynamic behaviors of the respective
LJ model in this region. For extracting the contribution
of the attraction part quantitatively, we calculate the fragility
π‘š for systems k1~ k6 at 𝜌 = 1.2, respectively. The results are
shown in Fig. 5(c). In RLJ systems, we can make the same
conclusion in RI as in the colloidal experiment15 that softness
makes strong glasses. In LJ systems, the contribution of
attraction produces a reversal correlation between k and π‘š,
implying that hardness makes strong glass, which is
4
FIG. 7. Angell plot of relaxation time 𝜏 versus scaled
reciprocal temperature Tg /T at volume fraction Ο• = 0.8.
(a): Comparison between four purely repulsive coresoftened systems ( rc = 1.0 ) in ascending order by
softness of repulsion with Ξ± = 1.5, 2.0, 2.5 and 5.0. (b):
Comparison between systems for Ξ± = 2.0 with attraction
(rc = 1.2) and pure repulsion (rc = 1.0). Through data
points in panels (a) and (b) are VFT fit lines.
FIG. 6. Schematic illustration of the scaled pair coresoftened potential 𝑉(π‘Ÿij )/Ο΅ for 𝛼 = 1.5, 2.0, 2.5 and 5.0,
respectively. The dashed line marks the position of
π‘Ÿij /𝜎ij = 1 which separates attractive from repulsive part
of the interaction potential.
consistent with recent studies.16-19However, the attraction
makes huge contribution in RI, so we can’t simply attribute
the different dynamic behaviors between LJ models to the
steepness of the repulsive part of interaction. The influence of
attraction weakened as k enhanced (or 𝜌 increased), which
suggests that the profile of PEL is controlled by repulsive part
as the repulsive force becomes the dominated interaction.
ratio of the mixture is set to be 1.4 to avoid
crystallization , and the detailed parameters related to
units setting for core-softened potential can be found in
Ref. [8]. The interaction in core-softened potential
between particles i and j is 𝑉(π‘Ÿπ‘–π‘— ) =
πœ–
𝛼
(1 βˆ’
π‘Ÿπ‘–π‘—
𝜎ij
Ξ±
) when
their separation π‘Ÿij is smaller than π‘Ÿc 𝜎ij with 𝜎ij being the
sum of their radii, πœ– the energy scale and π‘Ÿc the potential
cutoff, and zero otherwise. π‘Ÿc is chosen to be 1.0 to
mimic purely repulsive systems and we set π‘Ÿc to be a
value larger than 1.0 to introduce nonperturbative
attraction, e.g. π‘Ÿc = 1.2 in system with Ξ± = 2.0 in this
study. We study four systems with Ξ± = 1.5, 2.0, 2.5 and
5.0, respectively, at the same volume fraction Ο• defined
3
3
as πœ™ = (βˆ‘N
i=1 πœŽπ‘– )/𝐿 . The schematic plots of the four
core-softened potentials with the absence and presence of
attraction are shown in Fig. 6. Seen from Fig. 6 the
curves get steeper with decreasing Ξ±. Thus, the parameter
𝛼 can be used to tune the softness of interaction, i.e.
larger value of 𝛼 is indicative of softer interaction.
Figure 7 (a) compares Angell plot between the four
purely repulsive core-softened potential systems with
different softness of repulsion. The curve becomes more
flat with larger value of Ξ± (softer repulsion), suggesting
that softness makes strong glass formers, which is the
major conclusion about the softness dependence of
fragility in previous experimental15 and our theoretical
studies. It should be noted that we also get the similar
conclusion at other volume fractions above or below
jamming transition point 6 in purely repulsive coresoftened potential systems.
We can’t get models with arbitrary small softness in the
generalized Lennard-Jones potential as long as we keep
the exponents in the potential positive to make the
potential possess physical meaning. It is found that the
variance of the fragility in Fig. 3 (a) or (b) is small
compared with the experiment in Ref. [15]. However, the
It has been shown that structures of inherent structures
exhibit difference between RLJ and LJ systems at fixed
density when attraction in LJ system doesn’t act as
perturbation.24 We find the difference level of attraction
influence on fragility can also be estimated by the
attraction effect within the structural properties of zerotemperature glasses. Fig. 5 (d) presents the structural
difference between RLJ and LJ systems, which is
defined by βˆ†gAA(r) = gAA,rlj(r) βˆ’ gAA,lj(r), to quantify the
extent of attraction influence in LJ system. Obviously,
the attraction is perturbative in LJ system when βˆ†gAA(r)
= 0 for any distance r (such as in model D), and bigger
βˆ†gAA(r) reflects more important role of attraction (such
as in models A, B and C). Thus, the extent of attraction
influence on structure correlates well with that of
fragility difference between RLJ and LJ systems in
models A, B, C, and D, respectively, as shown in Fig. 5
(a), (b) and (d).
IV. FURTHER DISCUSSION IN CORE-SOFTENED
POTENTIAL SYSTEMS
Meanwhile, to further testify our conclusions drawn from
Lennard-Jonnes systems, we extend our research on
softness dependence of fragility to the widely used coresoftened potential system mimicking colloids with a
binary (50 : 50) mixture of N = 1000 frictionless spheres
having the same mass.8,33,36 By convention, the diameter
5
Kob-Anderson parameters. 26 We will enter at lower
densities the scenario of jamming transition or hardsphere glass transition in repulsive systems 3,6,35,36 and the
territory of inhomogeneous liquids or gels with formation
of cavitation in attractive systems,18,19,37 which are
beyond the scope of this study. It will also be interesting
to investigate systematically how the varying repulsive
steepness of interaction affects fragility when the
strength of attraction is held fixed as studied in Ref. 38,
and how the varying length and strength of attraction
influence fragility with the repulsive steepness of
interaction unchanged as done in Ref. 39. We also note
that the scaling collapse of g(r) measured at three quite
high pressures, 2.5, 18.2 and 31.4 GPa, is presented in a
recent experiment study40 on Pd81Si19 glassy alloy whose
pairwise interaction approximates Lennard-Jonnes
potential. Therefore, if possible, the high-pressure
experiment may be a good choice to verify our finding of
the invariant fragility above πœŒπ‘ for specific interatomic
potential. Furthermore, density, repulsion and attraction,
when mixed together, can make a variety of changes of
fragility, which will be an optional platform to test the
generality of the known correlations between fragility
and other parameters, such as Poisson’s ratio14 and
anharmonicity.16,17
core-softened potential whose softness of repulsion can
be tuned to be arbitrarily small, can approximate some
experimental colloidal systems,33 which is also one of
our purposes to include the discussion of the coresoftened potential systems here. In core-softened
systems, we can see nearly as large variance of fragility
in Fig. 7 (a) as observed in the experimental study,15
though
the interaction potential between colloidal
particles in Ref. [15] are not the same as our coresoftened potential. It should be mentioned that the
steepness of core-softened potential can't be tuned to be
arbitrary hardness due to its expression, and hence we
can’t observe the invariant fragility (only existing at very
hard steepness systems based on the study from repulsive
Lennard-Jonnes systems) at fixed volume fraction within
the steepness range we can get access to in repulsive
core-softened systems. Therefore, it seems that repulsive
core-softened and repulsive Lennard-Jonnes potentials
are complementary in studying the softness dependence
of fragility.
Figure 7 (b) compares Angell plot between the coresoftened potential (Ξ± =2.0) systems with the absence and
presence of attraction. As demonstrated in it, the
attraction within core-softened potential systems induces
enhancement of fragility, which suggests that attraction
makes fragile glasses is the general conclusion applying
to different potential systems. Though larger π‘š in system
with attraction may be related to the finding that
attractive system is more heterogeneous in dynamics than
its counterpart without attraction,23,34 more convincingly
microscopic explanation needs to be further explored.
ACKNOWLEDGEMENTS
We thank N. Xu and L. M. Xu for useful discussions. We
also acknowledge the computational support from the
Beijing Computational Science Research Center (CSRC).
The work was supported by the NSF of China (Grant No.
51571111) and the MOST 973 Program (No.
2015CB856800).
V. CONCLUSIONS
In this study, through MD simulations and including the
effect of attraction, we provided a qualitative picture of
how interaction potential and density determine fragility
of glass forming liquids, and unified the long-standing
inconsistence pertaining to the repulsive softness
dependence of fragility. In purely repulsive LennardJonnes systems, the fragility in the 𝜌 – k field can be
divided into three regions qualitatively: RI which can be
described by β€˜softness makes strong glasses’, RII where
fragility can only be tuned by 𝜌, and RIII with constant
fragility. It suggests the density plays an unexpected role
as we discuss the repulsive steepness dependence of
fragility. The level of the influence of attraction on
fragility can be estimated from the structure information
of related zero-temperature glasses, and a glass former
will be tuned more fragile if nonperturbative attraction is
added, especially in RI. However, due to the diverse
correlations between the fragility, density and atomic
interaction within different regions, to simply claim
either repulsion or attraction determines fragility is not
all-inclusive and sometimes may give rise to
inconclusive studies.
Our findings in the course of this study may point out
directions for further research. Our simulations are
mainly conducted at densities no less than ρ = 1.2 within
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